Operator algebras of functions

Abstract

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite-dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.

Interpolation and Transfer-function Realization for the Noncommutative Schur–Agler Class

“…(2) There is an auxiliary Hilbert space X so that the kernel on Ω given by This theorem follows by pasting together various pieces from the results of [9,17,18] (see also [60] for a somewhat different setting and point of view). The seminal work of Agler [1,2] handled the special case where Q is taken to be Q diag (z) := z 1 .…”

Universal commutative operator algebras and transfer function realizations of polynomials

“…However the proof of Theorem 1.2 shows that UC(E) is indeed determined by its finite-dimensional representations. (In other words, the operator algebra UC(E) is residually finite dimensional in the sense of [9]; this fact could also be deduced from their results).…”

“…As noted earlier, if the S j are matrices, then this is just the condition that S lies in the unit ball of E * . The algebras we are considering fall under the class considered in [9], as operator algebras of functions over the "quantized domain" ball(E * ). However here we take a somewhat different point of view, emphasizing the role of operator space duality, and the question of whether a given operator algebra of functions (such as the bounded analytic functions on the polydisk) is realizable by the present construction.…”

“…In our notation (up to taking closures) the algebra (P n , · u ) is UC(MAX( 1 n )). The above example shows that the norm on UC(MIN( 1 n )) strictly dominates the sup norm when n ≥ 3; it follows that none of the algebras It is also possible to describe the UC(E) algebras in terms of operator-valued analytic functions that act as bounded multipliers of a canonically determined family of reproducing kernels determined by E; in particular this description can be extracted from [9]. At this point, however, we do not know if this family of kernels has a more concrete description.…”

Residually finite-dimensional operator algebras

“…Studying this property in the setting of non-selfadjoint operator algebras is the driving force of this paper. Similar investigations can be found scattered in the literature (see for instance [30] and [16]), but we adopt here a somehow more systematic approach.…”